An investigation of certain characteristic properties of the

exponential distribution based on maxima in small samples

Barry C. Arnold

University of California, Riverside

Joint work with

Jose A. Villasenor

Colegio de Postgraduados

Montecillo, Mexico

As a change of pace, instead of looking at maxima of large samples.

As a change of pace, instead of looking at maxima of large samples.

Let’s look at smaller samples.

As a change of pace, instead of looking at maxima of large samples.

Let’s look at smaller samples.

Really small samples !!

As a change of pace, instead of looking at maxima of large samples.

Let’s look at smaller samples.

Really small samples !!

In fact n=2.

First, we note that neither (A*) nor (A**) is sufficient to guarantee that the X’s are exponential r.v.s.

For geometrically distributed X’s, (A*) and (A**) both hold, since the corresponding spacings are independent.

An obvious result

v

So, we have

Weibull distributions provide examples in which the covariance between the first two spacings is positive, negative or zero (in the exponential case).

But we seek an example in which we have zero covariance for a non-exponential distribution.

It’s not completely trivial to achieve this.

Power function distributions

Power function distributions

In this case we find:

Pareto (II) distributions

Pareto (II) distributions

Here the covariance is always positive

Open question

Does reciprocation always reverse the sign of the covariance ?

The hunt for a non-exponential example with zero covariance continues.

The hunt for a non-exponential example with zero covariance continues.

What would you try ?

The hunt for a non-exponential example with zero covariance continues.

What would you try ?

Success is just around the corner, or rather on the next slide.

Pareto (IV) or Burr distributions

Pareto (IV) or Burr distributions

So that

Pareto (IV) or Burr distributions

Can you find a “nicer” example ?

Extensions for n>2

Some negative results extend readily:

Back to Property (B)

Back to Property (B)

Recall:

This holds if the X’s are i.i.d. exponential r.v.’s. It is unlikely to hold for other parent distributions. More on this later.

Another exponential property

If a r.v. has a standard exponential distribution (with mean 1) then its density and its survival function are identical, thus

And it is well-known that property (C) only holds for the standard exponential distribution.

Combining (B) and (C).

By taking various combinations of (B) and (C) we can produce a long list of unusual distributional properties, that do hold for exponential variables and are unlikely to hold for other distributions.

Combining (B) and (C).

By taking various combinations of (B) and (C) we can produce a long list of unusual distributional properties, that do hold for exponential variables and are unlikely to hold for other distributions.

In fact we’ll list 10 of them !!

Combining (B) and (C).

By taking various combinations of (B) and (C) we can produce a long list of unusual distributional properties, that do hold for exponential variables and are unlikely to hold for other distributions.

In fact we’ll list 10 of them !!

Each one will yield an exponential characterization.

Combining (B) and (C).

By taking various combinations of (B) and (C) we can produce a long list of unusual distributional properties, that do hold for exponential variables and are unlikely to hold for other distributions.

In fact we’ll list 10 of them !!

Each one will yield an exponential characterization.

They appear to be closely related, but no one of them implies any other one.

Combining (B) and (C).

The good news is that I don’t plan to prove

or even sketch the proofs of all 10.

We’ll just consider a sample of them

The 10 characteristic properties

The 10 characteristic properties

The following 10 properties all hold if the X’s are standard exponential r.v.’s.

The 10 characteristic properties

The 10 characteristic properties

Property (2)

Property (2)PROOF:

Define

then

Property (2)PROOF continued:

and we conclude that

Property (5)

Property (5)PROOF:

From (5)

Property (5)

PROOF continued:

As before define

It follows that

We can write

and

Property (5)

PROOF continued:

which implies that

which implies that

Property (5)

PROOF continued:

For k>2 we have

which via induction yields for k>2.

So and

Property (10)

Property (10)

PROOF: Since

we have

and so

Property (10)PROOF continued:

also

Property (10)

PROOF continued:

But

so we have for every x,

i.e., a constant failure rate =1, corresponding to a standard exponential distribution.

Since we have lots of time, we can also go through the remaining 7 proofs.

Since we have lots of time, we can also go through the remaining 7 proofs.

HE CAN’T BE SERIOUS !!!

Thank you for your attention

Thank you for your attention

and for suffering through 3 of the 10 proofs !